Apparatus and method for minimizing accumulated rounding errors in coefficient values in a lookup table for interpolating polynomials

ABSTRACT

An apparatus and method are disclosed for minimizing accumulated rounding errors in coefficient values in a lookup table for interpolating polynomials. Unlike prior art methods that individually round each polynomial coefficient of a function, the method of the present invention use a “ripple carry” rounding method to round each coefficient using information from the previously rounded coefficient. The “ripple carry” method generates rounded coefficients that significantly improve the total rounding error for the function.

CROSS REFERENCE TO RELATED APPLICATION

The present invention is related to that disclosed in the following United States Non-Provisional Patent Application:

U.S. patent application Ser. No. 10/107,598, filed concurrently herewith on Mar. 26, 2002, entitled “APPARATUS AND METHOD FOR PROVIDING HIGHER RADIX REDUNDANT DIGIT LOOKUP TABLES FOR RECODING AND COMPRESSING FUNCTION VALUES.”

The above patent application is commonly assigned to the assignee of the present invention. The disclosures in this related patent application are hereby incorporated by reference for all purposes as if fully set forth herein.

FIELD OF THE INVENTION

The present invention relates generally to the field of computer technology. The present invention provides an improved apparatus and method for minimizing accumulated rounding errors in coefficient values in a lookup table for interpolating polynomials.

BACKGROUND OF THE INVENTION

In binary computing devices hardware direct lookup tables are typically employed for function evaluation and for reciprocal and root reciprocal seed values for division and square root procedures. For direct table lookup of a function of a normalized “p” bit argument 1≦x=1.b₁b₂ . . . b_(i)b_(i+1) . . . b_(p-1)<2, the “i” leading bits b₁b₂ . . . b_(i) provide an index to a table yielding “j” output bits that determine the approximate function value.

The calculation of values of elementary functions usually uses a polynomial approximation method. The accuracy of the coefficients of the polynomial determines the accuracy of the calculated value of the function. The polynomial coefficients are usually stored in a “constant store” portion (or lookup table) of a “read only memory” of an arithmetic logic unit of a data processor.

The accuracy of the each polynomial coefficient depends upon the number of bits used to express the polynomial coefficient. In practice the last digit of each polynomial coefficient is rounded to give an approximate value of the coefficient.

Prior art methods separately round each individual polynomial coefficient of a function. Because each individual polynomial coefficients is rounded separately, rounding errors accumulate and contribute to value of the total rounded error of the function.

Accordingly, there is a need in the art for a method of rounding the polynomial coefficients of a function so that rounding errors are minimized for each polynomial coefficient of the function. There is also a need in the art for a method of rounding the polynomial coefficients of a function so that total accumulated rounding errors are minimized for the function.

SUMMARY OF THE INVENTION

The present invention is directed to an apparatus and method for minimizing accumulated rounding errors in coefficient values in a lookup table for interpolating polynomials.

An advantageous embodiment of the present invention comprises an apparatus and method for minimizing accumulated rounding errors in coefficient values in a lookup table for interpolating polynomials. Unlike prior art methods that individually round each polynomial coefficient of a function, the method of the present invention use a “ripple carry” rounding method to round each coefficient using information from the previously rounded coefficient. The “ripple carry” method generates rounded coefficients that significantly improve the total rounding error for the function.

It is an object of the present invention to provide an apparatus and method for rounding the polynomial coefficients of a function so that rounding errors are minimized for each polynomial coefficient of the function.

It is another object of the present invention to provide an apparatus and method for rounding the polynomial coefficients of a function so that total accumulated rounding errors are minimized for the function.

It is also an object of the present invention to provide a data processor that contains rounded polynomial coefficients with minimum rounding errors.

The foregoing has outlined rather broadly the features and technical advantages of the present invention so that those skilled in the art may better understand the Detailed Description of the Invention that follows. Additional features and advantages of the invention will be described hereinafter that form the subject matter of the claims of the invention. Those skilled in the art should appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form.

Before undertaking the Detailed Description of the Invention, it may be advantageous to set forth definitions of certain words and phrases used throughout this patent document: The terms “include” and “comprise” and derivatives thereof, mean inclusion without limitation, the term “or” is inclusive, meaning “and/or”; the phrases “associated with” and “associated therewith,” as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, to bound to or with, have, have a property of, or the like; and the term “controller,” “processor,” or “apparatus” means any device, system or part thereof that controls at least one operation. Such a device may be implemented in hardware, firmware or software, or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document. Those of ordinary skill should understand that in many instances (if not in most instances), such definitions apply to prior uses, as well as to future uses, of such defined words and phrases.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taking in conjunction with the accompanying drawings, wherein like numbers designate like objects, and in which:

FIG. 1 illustrates a block diagram of a portion of a central processing unit showing a prior art arithmetic logic unit comprising a lookup table and an arithmetic unit.

FIG. 2 illustrates a block diagram of a prior art arithmetic logic unit comprising a lookup table, a multiplier recoder, and a multiplier unit;

FIG. 3 illustrates five equations that show how a prior art rounding method may be applied to round polynomial coefficients in a specific example;

FIG. 4 illustrates four equations that show how a prior art rounding method may be applied to round a first polynomial coefficient of a plurality of polynomial coefficients;

FIG. 5 illustrates four equations that show how a prior art rounding method may be applied to round a plurality of polynomial coefficients;

FIG. 6 illustrates four equations that show how the “ripple carry” rounding method of the present invention may be applied to round a plurality of polynomial coefficients;

FIG. 7 illustrates eight equations that show how the “ripple carry” rounding method of the present invention may be applied to round a plurality of polynomial coefficients for the function f(x) shown in Equation 8 of FIG. 3;

FIG. 8 illustrates three equations that show how the total rounding error may be calculated for the “ripple carry” rounding method of the present invention applied to the function f(x) shown in Equation 8 of FIG. 3;

FIG. 9 illustrates two equations that show how a polynomial expression for a function f(x) may be expressed with a plurality of polynomial coefficients where each polynomial coefficient is accurate to an i^(th) bit;

FIG. 10 illustrates nine equations that show how a total rounding error for the “ripple carry” method of the present invention may be expressed as a convex combination of weighted individual “ripple carry” rounding errors;

FIG. 11 illustrates a data processor comprising a memory comprising a constant store portion that contains polynomial coefficients rounded by the “ripple carry” rounding method of the present invention;

FIG. 12 illustrates a data processor comprising an arithmetic logic unit comprising a constant store read only memory (ROM) unit that contains polynomial coefficients rounded by the “ripple carry” rounding method of the present invention; and

FIG. 13 illustrates a flow chart showing the steps of one advantageous embodiment of a method of the present invention.

FIG. 14 illustrates a flow diagram in accordance with the present disclosure.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1 through 13, discussed below, and the various embodiments used to describe the principles of the present invention in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the invention. Those skilled in the art will understand that the principles of the present invention may be implemented in any suitably arranged lookup table unit in a central processing unit of a computer system.

FIG. 1 illustrates a block diagram of a portion of a central processing unit 100 showing a prior art arithmetic logic unit (ALU) 110. Arithmetic logic unit 110 comprises a lookup table (LUT) 120 and an arithmetic unit 130. Lookup table 120 receives data in the form of “i” input bits. Lookup table 120 outputs data to arithmetic unit 130 in the form of “j” output bits in accordance with principles that are well known in the prior art. Arithmetic unit 130 may comprise a multiplier unit, an adder unit, a microprogram storage unit, or other type of computational unit.

FIG. 2 illustrates a block diagram of a prior art arithmetic logic unit (ALU) 200 comprising lookup table 210, multiplier recoder 220, and multiplier unit 230. Lookup table 210 receives data in the form of “i” input bits. Lookup table 210 outputs data to multiplier recoder 220 in the form of “j” output bits in accordance with principles that are well known in the prior art. Multiplier recoder 220 comprises a Booth “radix 4” multiplier recoder. Also in accordance with principles that are well known in the prior art multiplier recoder 220 outputs to multiplier unit 230 a number of bits equal to $3\;\left\lceil \frac{\left( {j + 1} \right)}{2} \right\rceil$ bits where the bottomless brackets denote taking the smallest integer greater than or equal to the expression within the bottomless brackets.

The calculation of values of elementary functions is usually made using a polynomial approximation method. The accuracy of the coefficients of the polynomial determines the accuracy of the calculated value of the function. Consider the following polynomial approximation for the decimal logarithm of x. log₁₀ x=a ₁ t+a ₃ t ³ +a ₅ t ⁵ +a ₇ t ⁷ +a ₉ t ⁹ +E(x)  (1) where t=(x−1)/(x+1). The polynomial approximation is valid for x values in the range $\begin{matrix} {\frac{1}{\sqrt{10}} \leq x \leq \sqrt{10}} & (2) \end{matrix}$

The values of the polynomial coefficients are: a₁=0.86859 1718  (3) a₃=0.28933 5524  (4) a₅=0.17752 2071  (5) a₇=0.09437 6476  (6) a₉=0.19133 7714  (7)

The accuracy of the values of the polynomial coefficients shown in Equation 3 through Equation 7 is sufficient to ensure that the absolute value of the error E(x) is less than 10⁻⁷.

The values of the polynomial coefficients are stored in a hardware direct lookup table. The values of the polynomial coefficients must be rounded (or truncated) at some point for storage in a lookup table. The prior art method for rounding polynomial coefficients individually rounds each polynomial coefficient.

FIG. 3 shows five (5) equations that illustrate an exemplary prior art method for rounding coefficients in a polynomial that expresses the value of an exemplary function f(x). The function f(x) shown in Equation 8 of FIG. 1 is an even function. That is, the terms of the function f(x) are in even powers of x. Functions that are of odd powers of x and mixed powers of x will be discussed later.

The first coefficient of the function f(x) shown in FIG. 1 has a value of 0.14. This first coefficient will be designated with the expression C₀. The second coefficient (for x²) has a value of 0.34 and is designated with the expression C₂. The third coefficient (for x⁴) has a value of 0.74 and is designated with the expression C₄. Lastly, the fourth coefficient (for x⁶) has a value of 0.44 and is designated with the expression C₆.

For purposes of illustration the coefficients of f(x) will be assumed to be exact to two decimal places. Therefore, when the value of x is set equal to one (“1”) the exact value of the function f(1) shown in Equation 9 is 1.66.

The prior art method of rounding the coefficients in a polynomial comprises rounding each coefficient individually. The expression f_(a)(x) shown in Equation 10 of FIG. 3 shows the result of rounding the coefficients C₀ through C₆ in Equation 8. The value 0.14 of coefficient C₀ rounds down to the value 0.1. The value 0.34 of coefficient C₂ rounds down to the value 0.3. The value 0.74 of coefficient C₄ rounds down to the value 0.7. The value 0.44 of coefficient C₆ rounds down to the value 0.4.

In order to measure the accuracy provided by the expression f_(a)(x) with rounded coefficients the value of x is set equal to one (“1”) in Equation 10. The result is f_(a)(1) as shown in Equation 11. The value of f_(a)(1) is 1.5. Therefore, the error between the exact value of 1.66 for f(1) and the rounded value of 1.5 for f_(a)(1) is 0.16. This result is shown in Equation 12. This error value of 0.16 will later be compared with an error value obtained using the method of the present invention.

In Equation 13 of FIG. 4 the function f(x) is shown with the coefficients expressed as C₀, C₂, C₄ and C₆. In Equation 14 the function f_(a)(x) is shown with the rounded coefficients expressed as A₀, A₂, A₄ and A₆. In the prior art method of rounding coefficients each rounded coefficient A_(i) is obtained by rounding its corresponding coefficient C_(i). An example of this is shown in Equation 15 where the rounded coefficient A₀ is equal to the rounded value of coefficient C₀.

The rounding error E_(i) for each coefficient is obtained by subtracting the value of the rounded coefficient A_(i) from the value of its corresponding coefficient C_(i). An example of this is shown in Equation 16 where the rounding error E₀ is equal to the value of the coefficient C₀ minus the value of the rounded coefficient A₀.

FIG. 5 generally illustrates the equations that are used in the prior art method of rounding coefficients. As shown in Equations 17 through 20, each rounded coefficient A₁ is obtained by rounding its corresponding coefficient C_(i). The rounding error E_(i) for each coefficient is obtained by subtracting the value of the rounded coefficient A_(i) from the value of its corresponding coefficient C_(i) For example, rounded coefficient A₄ is obtained by rounding the corresponding coefficient C₄. The rounding error E₄ for rounded coefficient A₄ is obtained by subtracting the value of the rounded coefficient A₄ from the value of corresponding coefficient C₄.

FIG. 6 illustrates the method of rounding coefficients of the present invention. Only the first rounded coefficient A₀ is obtained using the coefficient rounding method of the prior art. This step is shown in Equation 21. In the next portion of the method of present invention (shown in Equation 22), the rounded coefficient A₂ is obtained by adding the rounding error E₀ (from Equation 21) to coefficient C₂ and rounding the sum of C₂ and E₀. The rounding error E₂ for rounded coefficient A₂ is then computed by subtracting the value of rounded coefficient A₂ from the sum of coefficient C₂ and rounding error E₀ (as shown in Equation 22).

In the next portion of the method of present invention (shown in Equation 23), the rounded coefficient A₄ is obtained by adding the rounding error E₂ (from Equation 22) to coefficient C₄ and rounding the sum of C₄ and E₂. The rounding error E₄ for rounded coefficient A₄ is then computed by subtracting the value of rounded coefficient A₄ from the sum of coefficient C₄ and rounding error E₂ (as shown in Equation 23).

In the next portion of the method of present invention (shown in Equation 24), the rounded coefficient A₆ is obtained by adding the rounding error E₄ (from Equation 23) to coefficient C₆ and rounding the sum of C₆ and E₄. The rounding error E₆ for rounded coefficient A₆ is then computed by subtracting the value of rounded coefficient A₆ from the sum of coefficient C₆ and rounding error E₄ (as shown in Equation 24).

The method of the present invention can be continued for as many terms as there are in the polynomial. In the present example there are only four terms in the function f(x) and so only four rounded coefficients (A₀, A₂, A₄, A₆) are calculated. Except for the first rounded coefficient A₀, the method of the present invention calculates each rounded coefficient A_(i) using the value of the rounding error for the previous rounded coefficient. For this reason the method of the present invention is referred to as the “ripple carry” rounding method.

FIG. 7 illustrates how the “ripple carry” rounding method of the present invention may be applied to calculate rounded coefficients for the function f(x) shown in Equation 8 of FIG. 3. The first rounded coefficient A₀ is obtained from rounding the value 0.14 of C₀. As shown in Equation 25 the result for rounded coefficient A₀ is 0.1. The rounding error E₀ is equal to C₀ minus A₀. This value is 0.14 minus 0.1. As shown in Equation 26 the result for rounding error E₀ is 0.04.

The next rounded coefficient A₂ is obtained from rounding the sum of C₂ (C₂ equals 0.34) and the rounding error E₀ (E₀ equals 0.04). The sum of C₂ and E₀ is 0.38. As shown in Equation 27 the result for rounded coefficient A₂ is 0.4. The rounding error E₂ is equal to C₂ plus E₀ minus A₂. This value is 0.34 plus 0.04 minus 0.4. As shown in Equation 28 the result for rounding error E₂ is negative 0.02.

The next rounded coefficient A₄ is obtained from rounding the sum of C₄ (C₄ equals 0.74) and the rounding error E₂ (E₂ equals negative 0.02). The sum of C₄ and E₂ is 0.72. As shown in Equation 29 the result for rounded coefficient A₄ is 0.7. The rounding error E₄ is equal to C₄ plus E₂ minus A₄. This value is 0.74 plus (negative 0.02) minus 0.7. As shown in Equation 30 the result for rounding error E₂ is a positive 0.02.

The next rounded coefficient A₆ is obtained from rounding the sum of C₆ (C₆ equals 0.44) and the rounding error E₄ (E₄ equals 0.02). The sum of C₆ and E₄ is 0.46. As shown in Equation 31 the result for rounded coefficient A₆ is 0.5. The rounding error E₆ is equal to C₆ plus E₄ minus A₆. This value is 0.44 plus 0.02 minus 0.5. As shown in Equation 32 the result for rounding error E₆ is a negative 0.04.

FIG. 8 illustrates how the rounded coefficient values A₀, A₂, A₄, and A₆ generated by the “ripple carry” rounding method described in FIG. 7 may be applied to calculate an approximate value of the function f(x) shown in Equation 8 of FIG. 3. Inserting the rounded coefficient values A₀, A₂, A₄, and A₆ into Equation 14 of FIG. 4 gives Equation 33 of FIG. 8. Equation 33 is a more accurate approximation of Equation 8 than the approximation of Equation 10 that was obtained by the prior art rounding method.

The increase in accuracy provided by the method of the present invention may be seen by calculating the value of f_(a)(x) in Equation 33 for the value of x equal to one (“1”). As shown in Equation 34, f_(a)(1) equals 1.7. Therefore, the error between the exact value of 1.66 for f(1) and the rounded value of 1.7 for f_(a)(1) in Equation 34 is a negative 0.04. This result is shown in Equation 35. The absolute value of the error obtained using the “ripple carry” rounding method of the present invention is four (4) times more accurate than the value of the error obtained using the prior art method of rounding.

This relatively simple example set forth above provides an understanding how the “ripple carry” rounding method of the present invention operates. In practice many digits are used to express each coefficient in a high level of accuracy. For example, it is not unusual for a coefficient term to be represented as many as sixty four (64) binary digits.

Consider a transcendental function f(x) over a normalized interval 1≦x≦2. Suppose that a value of x is represented by “p” digits in the form: x=1.b₁b₂b₃ . . . b_(i)b_(i+1) . . . b_(p-1)  (36) where the integer one (“1”) is represented by one bit and the fraction is represented by “p-1” binary bits (i.e., bits b₁ through b_(p-1)). A value of x that is truncated at bit b_(i) is designated with the symbol x_(i). x_(i)=1.b₁b₂b₃ . . . b_(i)  (37) Then let the letter “d” designate a fraction that is represented by bits b₁₊₁ through bit b_(p-1). d=0.b₁₊₁b_(i+2)b₁₊₃ . . . b_(p-1)  (38) The expression for x in Equation 36 may then be represented by x=x _(i) +d2^(−i)  (39) where the fraction “d” has been multiplied by the factor 2^(−i) to reduce the value of “d” to an appropriate value so that the value “d 2^(−i)” added to x_(i) yields the value x.

Now suppose that the function f(x) is approximated by an even polynomial shown in Equation 40 of FIG. 9. The term “FE” represents a “function error” that is of order “d⁸2^(−8i).” FE is suitably small and may be neglected. As shown in Equation 41, each of the coefficients of f(x) (C₀, C₂, C₄, and C₆) is represented by a value that is truncated at the i^(th) bit. That is, bit b_(i) is the last bit in the fraction of each coefficient.

Each coefficient of f(x) (C₀, C₂, C₄, and C₆) is determined by looking up a value in a lookup table. The lookup table is indexed by the same leading “i” bits of the normalized argument's fraction.

In a prior art lookup table each coefficient is truncated independently to provide an output for the lookup table. Each coefficient is truncated independently to the same last fixed point position designated with the letter “n.” This means that an independent rounding error of order 2^(−(n+1)) is introduced for each coefficient. When, as in our example, there are four coefficients, then four independent rounding errors of order 2^(−(n+1)) are introduced. In a worst case scenario when the fraction “d” approaches a value of one (“1”), the total rounding error could approach a value of four (4) times the value 2^(−(n+1)). That is, the total rounding error could approach a value of 4[2^(−(n+1))].

The “ripple carry” rounding method of the present invention solves this problem by iteratively creating lookup table values so that the total rounding error is a convex combination of the individual rounding errors. A “convex” combination is a linear combination that sums to a value of one (“1”). The total rounding error using the method of the present invention is bounded by the value 2^(−(n+1)). The “ripple carry” rounding method provides a significant improvement over the accuracy obtainable by prior art methods.

The 2^(−(n+1)) bound on the total rounding error of the “ripple carry” rounding method by seen by considering the “ripple carry” rounding equations. FIG. 10 illustrates several expressions for the total rounding error “f(x)−f_(a)(x).” Subtracting Equation 14 from Equation 13 yields Equation 42 of FIG. 10. Equation 43 of FIG. 10 is obtained by substituting the expressions for the rounding error values E₀ through E₆ from Equations 21 through 24. Equation 44 of FIG. 10 is obtained by factoring out the individual rounding error values E₀ through E₆.

As can be seen with reference to Equation 44 each rounding error value E₀ through E₆ may be considered to be multiplied by a “weight value” (w_(i) where i=0, 2, 4, 6) As shown in Equations 45 through 48, the weight value for E₀ is (1−x²), the weight value for E₂ is x²(1−x²), the weight value for E₄ is x⁴(1−x²), and the weight value for E₆ is x⁶. Therefore, the total rounding error may be expressed in Equation 49 as: f(x)−f _(a)(x)=E ₀ w ₀ +E ₂ w ₂ +E ₄ W ₄ +E ₆ w ₆  (49) The weight values, w_(i), satisfy the condition: 0≦w_(i)≦1 for 0≦x≦1  (50) The weight values, w_(i), also satisfy the condition: w ₀ +w ₂ +w ₄ +w ₆=1  (51) Equation 49 shows that the total rounding error for the “ripple carry” rounding method of the present invention is a convex combination of E₀ and E₂ and E₄ and E₆, the individual “ripple carry” rounding errors. Therefore, the maximum value of the total “ripple carry” rounding error is bounded (as shown in Equation 52 of FIG. 10) by the value 2^(−(n+1)).

FIG. 11 illustrates a data processor 1110 constructed in accordance with the principles of the present invention. Data processor 1110 comprises memory 1120 capable of containing at least one program 1130. Memory 1120 comprises a constant store portion 1140 that contains polynomial coefficients that have been rounded by the “ripple carry” rounding method of the present invention.

Memory 1120 may comprise random access memory (RAM) or a combination of random access memory (RAM) and read only memory (ROM). Memory 1120 may comprise a non-volatile random access memory (RAM), such as flash memory. In an alternate advantageous embodiment of data processor 1110, memory 1120 may comprise a mass storage data device, such as a hard disk drive (not shown). Memory 1120 may also include an attached peripheral drive or removable disk drives (whether embedded or attached) that reads read/write DVDs or re-writable CD-ROMs. As illustrated schematically in FIG. 11, removable disk drives of this type are capable of receiving and reading re-writable CD-ROM disk 1150.

FIG. 12 illustrates a data processor 1210 constructed in accordance with the principles of the present invention. Data processor 1210 comprises arithmetic logic unit 1220 capable of containing at least one microprogram 1230. Arithmetic logic unit 1220 comprises constant store unit 1240 that contains polynomial coefficients that have been rounded by the “ripple carry” rounding method of the present invention.

FIG. 13 illustrates a flow chart showing the steps of one advantageous embodiment of a method of the present invention. The steps are collectively referred to with reference numeral 1300. The first coefficient C₀ is rounded to obtain the first rounded coefficient A₀ (step 1310). Then the first rounding error E₀ is calculated using the equation E₀=C₀−A₀ (step 1220).

The index “j” is then set equal to one (“1”) (step 1330). Then the next rounded coefficient A_(2j) is calculated by rounding the sum (C_(2j)+E_(2j-2)) (step 1340). Then the next value of the rounding error E_(2j) is calculated by subtracting A_(2j) from the sum (C_(2j)+E_(2j-2)) (step 1350). The values A_(2j) and E_(2j) are stored in memory.

Then a determination is made whether the index “j” is equal to the maximum value of “j.” (step 1360). If the value of the index “j” is not equal to the maximum value of “j,” then the index “j” is incremented (step 1370). Control is then passed to step 1340 to calculate the next values of A_(2j) and E_(2j) for the new value of “j.”

When the value of the index “j” equals the maximum value of “j,” then all of the required values of A_(2j) and E_(2j) have been calculated. Control is then passed to the next portion of the computer software (not shown) that uses the calculated values of A_(2j) and E_(2j).

An advantageous method of the present invention has been described for an “even” function (i.e., a function of even powers of x such as x², x⁴, x⁶, etc.). The method of the present invention is equally applicable for an “odd” function (i.e., a function of odd powers of x such as x, x³, x⁵, x⁷, etc.). The index “2j” is replaced by the index “2j-1.”

The method of the present invention may also be applied to a function of “mixed” powers of x (i.e., a function of both even and odd powers of x). For a “mixed” powers function, the function is first separated into to its odd and even parts. The method is applied to the odd and even parts separately. The rounded odd coefficients are used for the odd powers of x in the function and the rounded even coefficients are used for the even powers of x in the function.

The total amount of rounding error of a “mixed” powers function is the sum of the rounding error of the “even” powers portion and the rounding error of the “odd” powers portion. Therefore, the total amount of rounding error for a “mixed” powers function is approximately twice the rounding error of either the “even” or the “odd” powers portion.

FIG. 14 illustrates a flow diagram in accordance with the present disclosure. At step 1410, a first set of rounded polynomial coefficients A2j of a function are obtained where j=1, 2, . . . , n for even powers of said function. At step 1420, a second set of rounded polynomial coefficients A2j-1 of said function are obtained where j=1, 2, . . . , n for even powers of said function.

The above examples and description have been provided only for the purpose of illustration, and are not intended to limit the invention in any way. As will be appreciated by the skilled person, the invention can be carried out in a great variety of ways, employing more than one technique from those described above, all without exceeding the scope of the invention. 

1. A method for obtaining a set of rounded polynomial coefficients of a function comprising: (a) obtaining a set of polynomial coefficients C_(2j) of a function where j=0, 1, 2, . . . , n; (b) rounding a first coefficient C_(2j) to obtain a first rounded coefficient A_(2j) for j=0; (c) calculating a first rounding error E_(2j) for j=0 by subtracting said first rounded coefficient A_(2j) for j=0 from said first coefficient C_(2j) for j=0; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j) by rounding the sum of coefficient C_(2j) and rounding error E_(2j-2); (f) calculating a value for next rounding error E_(2j) by subtracting a value of A_(2j) from the sum of coefficient C_(2j) and rounding error E_(2j-2); and repeating (d), (e), and (f) until the value of j exceeds a predetermined maximum value.
 2. A method as claimed in claim 1 wherein said function comprises a transcendental function.
 3. A method as claimed in claim 1 further comprising: storing a plurality of rounded coefficients A_(2j) in a computer-readable storage medium.
 4. A method as claimed in claim 3 wherein said computer-readable storage medium is a constant store read only memory in a data processor.
 5. A method as claimed in claim 1 further comprising: (a) obtaining a set of polynomial coefficients C_(2j-1) of a function where j=1, 2, . . . , n; (b) rounding a first coefficient C_(2j-1) to obtain a first rounded coefficient A_(2j-1) for j=1; (c) calculating a first rounding error E_(2j-1) for j=1 by subtracting said first rounded coefficient A_(2j-1) for j=1 from said first coefficient C_(2j-1) for j=1; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j-1) by rounding the sum of coefficient C_(2j-1) and rounding error E_(2j-3); (f) calculating a value for next rounding E_(2j-1) by subtracting a value of A_(2j-1) from the sum of coefficient C_(2j-1) and rounding error E_(2j-3); and repeating (d), (e), and (f) until the value of j exceeds a predetermined maximum value.
 6. A method as claimed in claim 5 wherein said function comprises at least one of: a trigonometric function, an exponential function, a logarithmic function, and a transcendental function.
 7. A method as claimed in claim 5 further comprising: storing a plurality of rounded coefficients A_(2j) in a computer-readable storage medium.
 8. A method as claimed in claim 7 wherein said computer-readable storage medium is a constant store read only memory in a data processor.
 9. A method as claimed in claim 1 wherein said function comprises a trigonometric function.
 10. A method as claimed in claim 1 wherein said function comprises an exponential function.
 11. A method as claimed in claim 1 wherein said function comprises a logarithmic function.
 12. A data processor comprising: a memory containing rounded polynomial coefficients created by (a) obtaining a set of polynomial coefficients C_(2j) of a function where j=0, 1, 2, . . . , n; (b) rounding a first coefficient C_(2j) to obtain a first rounded coefficient A_(2j) for j=0; (c) calculating a first rounding error E_(2j) for j=0 by subtracting said first rounded coefficient A_(2j) for j=0 from said first coefficient C_(2j) for j=0; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j) by rounding the sum of coefficient C_(2j) and rounding error E_(2j-2); (f) calculating a value for next rounding error E_(2j) by subtracting a value of A_(2j) from the sum of coefficient C_(2j) and rounding error E_(2j-2); and repeating (d), (e), and (f) until the value of j exceeds a predetermined maximum value.
 13. A data processor as claimed in claim 12 wherein obtaining the set of polynomial coefficients A_(2j-1) comprises a ripple carry rounding method.
 14. A data processor comprising: a memory containing rounded polynomial coefficients created by (a) obtaining a set of polynomial coefficients C_(2j-1) of a function where j=1, 2, . . . , n; (b) rounding a first coefficient C_(2j-1) to obtain a first rounded coefficient A_(2j-1) for j=1; (c) calculating a first rounding error E_(2j-1) for j=1 by subtracting said first rounded coefficient A_(2j-1) for j=1 from said first coefficient C_(2j-1) for j=1; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j-1) by rounding the sum of coefficient C_(2j-1) and rounding error E_(2j-3); (f) calculating a value for next rounding E_(2j-1) by subtracting a value of A_(2j-1) from the sum of coefficient C_(2j-1) and rounding error E_(2j-3); and repeating (d), (e), and (f) until the value of j exceeds a predetermined maximum value.
 15. A method for obtaining a set of rounded polynomial coefficients of a function comprising: (a) obtaining a first set of rounded polynomial coefficients A_(2j) of said function where j=1, 2, . . . , n for even powers of said function using a ripple carry rounding method; and (b) obtaining a second set of rounded polynomial coefficients A_(2j-1) of said function where j=1, 2, . . . , n for odd powers of said function.
 16. The method as claimed in claim 15 wherein obtaining said second set of rounded polynomial coefficients A_(2j-1) comprises a ripple carry rounding method.
 17. A method as claimed in claim 15 wherein said function comprises at least one of: a trigonometric function, an exponential function, a logarithmic function, and a transcendental function.
 18. A method as claimed in claim 15 further comprising: storing said first set of rounded polynomial coefficients A_(2j) and said second set of rounded polynomial coefficient A_(2j-1) in a computer-readable storage medium.
 19. A method as claimed in claim 18 wherein said computer-readable storage medium is a constant store read only memory in a data processor.
 20. For use in a data processor capable of storing polynomial coefficients of a function, computer-executable instructions stored on a computer-readable storage medium for rounding polynomial coefficient values, the computer-executable instructions comprising: (a) obtaining a set of polynomial coefficients C_(2j) of a function where j=0, 1, 2, . . . , n; (b) rounding a first coefficient C_(2j) to obtain a first rounded coefficient A_(2j) for j=0; (c) calculating a first rounding error E_(2j) for j=0 by subtracting said first rounded coefficient A_(2j) for j=0 from said first coefficient C_(2j) for j=0; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j) by rounding the sum of coefficient C_(2j) and rounding error E_(2j-2); (f) calculating a value for next rounding error E_(2j) by subtracting a value of A_(2j) from the sum of coefficient C_(2j) and rounding error E_(2j-2); and repeating(d), (e), and (f) until the value of j exceeds a predetermined maximum value.
 21. Computer-executable instructions stored on a computer-readable storage medium for rounding polynomial coefficient values as claimed in claim 19, wherein said computer-executable instructions comprise: (a) obtaining a set of polynomial coefficients C_(2j-1) of a function where j=1, 2, . . . , n; (b) rounding a first coefficient C_(2j-1) to obtain a first rounded coefficient A_(2j-1) for j=1; (c) calculating a first rounding error E_(2j-1) for j=1 by subtracting said first rounded coefficient A_(2j-1) for j=1 from said first coefficient C_(2j-1) for j=1; (d) incrementing the value of j by one; (e) calculating a value for next rounded coefficient A_(2j-1) by rounding the sum of coefficient C_(2j-1) and rounding error E_(2j-3); (f) calculating a value for next rounding error E_(2j-1) by subtracting a value of A_(2j-1) from the sum of coefficient C_(2j-1) and rounding error E_(2j-3); and repeating (d), (e), and (f) until the value of j exceeds a predetermined maximum value. 